Online, I was making the joke that Penultimate Day, defined as the day before New Year’s Eve, was the day each year when mathematicians and content creators on social media both tend to perk up each year, finding fun ways to express 2026, the incoming new year, from a mathematical/number theoretic perspective. The six ways to dissect a 2026x2026 square into 30 squares of squares (SPSS form) have so far taken the prize…
Common to all of Ed Pegg’s and Anton Antonov’s equations is that they draw from the six basic arithmetic operators, {+, –, ×, ÷, ^, √}, aside from ⌈cosh(9)/2⌉ == 2026, combining the hyperbolic cosine trig function with rounding. Is it a cheat to round? Maybe just a little, but when you consider just how vast a range cosh is sweeping through near nine, the fact that cosh(9)/2 is anywhere near 2026 is quite an amazing coincidence!!
To me, while the digit expressions running from 1 through 9, or 9 through 1 in Ed Pegg’s case are interesting, since these involve numbers greater than 26, they’re a bit less exciting. Again: From a totally aesthetic standpoint.
About 123/4!×56×7 + 8 + 9
This expression common to both Ed’s list of expressions combining the digits 1 to 9 (and also 9 to 1) with arithmetic operators, and Anton’s list combining the digits 9 to 1, is perhaps my favorite of the 1-9, 9-1 group, though violating my arbitrary aesthetic that none of the integers should be greater than 26. If it were crafted by a human strategy and not an exhaustive search, I would commend the creator of this expression for the daring move to go rational right off with 123/4! = 41/8. But then, in the very next ‘move’, multiplying by 56, the factors of 2 all cancel, and you are left with 287=7×41. Multiply by another 7, and you’re now at 2009=7×41, a composite number precisely 17 shy of where you need to be to hit 2026 precisely through the additions of 8 and 9, in which all bets are off in terms of prime factors. By adding 8, the prior prime factors encountered, 7 and 41, are replaced with new prime factors.
The penultimate number hit by adding 2009+8 = 2017 is perfect, because 2026 lies in the gap between the prime numbers 2017 and 2027. Not a record-breaking gap between primes, but 2017 is still a nice little prime number before the final hop by adding 9 to reach the 2026 destination.
I know it’s a bit ridiculous to analyze such an expression as if it were a nine-inning ballgame, but the multiplicative steps initially performed to get to 2009, and then the two additive steps to hit the prime below 2026 to reach 2026, is a very nice touch whether this is a hand-constructed expression and not found by brute search.
It’s also quite interesting to me that the two lists of 1-9 expressions only have this one expression in common, showing two radically different search algorithms that are obviously not exhaustive. Perhaps I’ll have a bit of time to cook up my own search algorithm to find •all• such expressions. And then also to see if there is •any• expression of this form which does not involve using a number greater than 26, meaning that only the numbers 12 or 23 would be allowed in addition to 1 through 9, or 21 when going from 9 to 1.
Interesting stuff.
About Ed’s two exponential-ish equations
Presented in the opposite direction as I intend to do by showing the two exponential functons, 2^x and cosh(x), I suspect readers in my STEAM group will be able to spot the √2²²-22 solution using the hint “Using only two distinct integers between 1 and 26 and only three arithmetic operators, construct an expression for 2026”, there will at least one person able to find that before it’s actually 2026. And that’s simply because one would note right off that 2¹¹=2048 is fairly close to the target. But then if I hint “Using cosh with two distinct integers in the range 1 through 26 and one operator NOT in the bag of six operators, can you find an expression for 2026?” that will stump most folks as they may tend not to think that if an answer is inexact, that “out of the box” operator might involve the ceiling function.
In any case, the content creator here picking up from the trail left by the math creators to create a nice little online puzzle. I appreciate all the runtime effort you guys put into finding those 1-9 and 9-1 expressions, as that must have taken a bit of CPU time.
And we’re now into New Year’s Eve in my timezone. Time to publish or perish. ;-)
